Entanglement Entropy in the Two-Dimensional Random Transverse Field Ising Model

The scaling behavior of the entanglement entropy in the two-dimensional random transverse field Ising model is studied numerically through the strong disordered renormalization group method. We find that the leading term of the entanglement entropy always scales linearly with the block size. However, besides this \emph{area law} contribution, we find a subleading logarithmic correction at the quantum critical point. This correction is discussed from the point of view of an underlying percolation transition, both at finite and at zero temperature.Comment: 4.3 pages, 4 figure

Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model

We establish a relation between several entanglement properties in the Lipkin-Meshkov-Glick model, which is a system of mutually interacting spins embedded in a magnetic field. We provide analytical proofs that the single-copy entanglement and the global geometric entanglement of the ground state close to and at criticality behave as the entanglement entropy. These results are in deep contrast to what is found in one- dimensional spin systems where these three entanglement measures behave differently.Comment: 4 pages, 2 figures, published versio

Area law and vacuum reordering in harmonic networks

We review a number of ideas related to area law scaling of the geometric entropy from the point of view of condensed matter, quantum field theory and quantum information. An explicit computation in arbitrary dimensions of the geometric entropy of the ground state of a discretized scalar free field theory shows the expected area law result. In this case, area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations. Furthermore, the explicit control on all the eigenvalues of the reduced density matrix allows for a verification of entropy loss along the renormalization group trajectory driven by the mass term. A further result of our computation shows that single-copy entanglement also obeys area law scaling, majorization relations and decreases along renormalization group flows.Comment: 15 pages, 6 figures; typos correcte

Entanglement entropy in one-dimensional disordered interacting system: The role of localization

The properties of the entanglement entropy (EE) in one-dimensional disordered interacting systems are studied. Anderson localization leaves a clear signature on the average EE, as it saturates on length scale exceeding the localization length. This is verified by numerically calculating the EE for an ensemble of disordered realizations using density matrix renormalization group (DMRG). A heuristic expression describing the dependence of the EE on the localization length, which takes into account finite size effects, is proposed. This is used to extract the localization length as function of the interaction strength. The localization length dependence on the interaction fits nicely with the expectations.Comment: 5 pages, 4 figures, accepted for publication in Physical Review Letter

Violation of area-law scaling for the entanglement entropy in spin 1/2 chains

Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half chain. Our result is complementary to the known relation between non-translational invariant, nearest neighbor interacting Hamiltonians and QMA complete problems.Comment: 9 pages, 4 figure

Optimal minimal measurements of mixed states

The optimal and minimal measuring strategy is obtained for a two-state system prepared in a mixed state with a probability given by any isotropic a priori distribution. We explicitly construct the specific optimal and minimal generalized measurements, which turn out to be independent of the a priori probability distribution, obtaining the best guesses for the unknown state as well as a closed expression for the maximal mean averaged fidelity. We do this for up to three copies of the unknown state in a way which leads to the generalization to any number of copies, which we then present and prove.Comment: 20 pages, no figure

Reduced density matrix and entanglement entropy of permutationally invariant quantum many-body systems

In this paper we discuss the properties of the reduced density matrix of quantum many body systems with permutational symmetry and present basic quantification of the entanglement in terms of the von Neumann (VNE), Renyi and Tsallis entropies. In particular, we show, on the specific example of the spin $1/2$ Heisenberg model, how the RDM acquires a block diagonal form with respect to the quantum number $k$ fixing the polarization in the subsystem conservation of $S_{z}$ and with respect to the irreducible representations of the $\mathbf{S_{n}}$ group. Analytical expression for the RDM elements and for the RDM spectrum are derived for states of arbitrary permutational symmetry and for arbitrary polarizations. The temperature dependence and scaling of the VNE across a finite temperature phase transition is discussed and the RDM moments and the R\'{e}nyi and Tsallis entropies calculated both for symmetric ground states of the Heisenberg chain and for maximally mixed states.Comment: Festschrift in honor of the 60th birthday of Professor Vladimir Korepin (11 pages, 5 figures

Tremor in motor neuron disease may be central rather than peripheral in origin

BACKGROUND AND PURPOSE: Motor neuron disease (MND) refers to a spectrum of degenerative diseases affecting motor neurons. Recent clinical and post-mortem observations have revealed considerable variability in the phenotype. Rhythmic involuntary oscillations of the hands during action, resembling tremor, can occur in MND, but their pathophysiology has not yet been investigated. METHODS: A total of 120 consecutive patients with MND were screened for tremor. Twelve patients with action tremor and no other movement disorders were found. Ten took part in the study. Tremor was recorded bilaterally using surface electromyography (EMG) and triaxial accelerometer, with and without a variable weight load. Power spectra of rectified EMG and accelerometric signal were calculated. To investigate a possible cerebellar involvement, eyeblink classic conditioning was performed in five patients. RESULTS: Action tremor was present in about 10% of our population. All patients showed distal postural tremor of low amplitude and constant frequency, bilateral with a small degree of asymmetry. Two also showed simple kinetic tremor. A peak at the EMG and accelerometric recordings ranging from 4 to 12 Hz was found in all patients. Loading did not change peak frequency in either the electromyographic or accelerometric power spectra. Compared with healthy volunteers, patients had a smaller number of conditioned responses during eyeblink classic conditioning. CONCLUSIONS: Our data suggest that patients with MND can present with action tremor of a central origin, possibly due to a cerebellar dysfunction. This evidence supports the novel idea of MND as a multisystem neurodegenerative disease and that action tremor can be part of this condition

Area laws for the entanglement entropy - a review

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: The entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such "area laws" for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium we review the current status of area laws in these fields. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation, and disordered systems, non-equilibrium situations, classical correlation concepts, and topological entanglement entropies are discussed. A significant proportion of the article is devoted to the quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. We discuss matrix-product states, higher-dimensional analogues, and states from entanglement renormalization and conclude by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations.Comment: 28 pages, 2 figures, final versio

Molecular evolution of aphids and their primary ( Buchnera sp.) and secondary endosymbionts: implications for the role of symbiosis in insect evolution.

Aphids maintain an obligate, endosymbiotic association with Buchnera sp., a bacterium closely related to Escherichia coli. Bacteria are housed in specialized cells of organ-like structures called bacteriomes in the hemocoel of the aphid and are maternally transmitted. Phylogenetic studies have shown that the association had a single origin, dated about 200-250 million years ago, and that host and endosymbiont lineages have evolved in parallel since then. However, the pattern of deepest branching within the aphid family remains unsolved, which thereby hampers tin appraisal of, for example, the role played by horizontal gene transfer in the early evolution of Buchnera. The main role of Buchnera in this association is the biosynthesis and provisioning of essential amino acids to its aphid host. Physiological and metabolic studies have recently substantiated such nutritional role. In addition, genetic studies of Buchnera from several aphids have shown additional modifications, such as strong genome reduction, high A+T content compared to free-living bacteria, differential evolutionary rates, a relative increase in the number of non-synonymous substitutions, and gene amplification mediated by plasmids. Symbiosis is an active process in insect evolution cis revealed by the intermediate values of the previous characteristics showed by secondary symbionts compared to free-living bacteria and Buchnera